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52 Monoids
52.1 Functions for Monoids
52.1-1 IsMonoid
52.1-2 Monoid
52.1-3 Submonoid
52.1-4 MonoidByGenerators
52.1-5 AsMonoid
52.1-6 AsSubmonoid
52.1-7 GeneratorsOfMonoid
52.1-8 TrivialSubmonoid
52.1-9 FreeMonoid
52.1-10 MonoidByMultiplicationTable

### 52 Monoids

This chapter describes functions for monoids. Currently there are only few of them. More general functions for magmas and semigroups can be found in Chapters 35 and 51.

#### 52.1 Functions for Monoids

##### 52.1-1 IsMonoid

`‣ IsMonoid`( D ) | ( property )

A monoid is a magma-with-one (see 35) with associative multiplication.

##### 52.1-2 Monoid

`‣ Monoid`( gen1, gen2, ... ) | ( function )

`‣ Monoid`( gens[, id] ) | ( function )

In the first form, `Monoid` returns the monoid generated by the arguments gen1, gen2, ..., that is, the closure of these elements under multiplication and taking the 0-th power. In the second form, `Monoid` returns the monoid generated by the elements in the homogeneous list gens; a square matrix as only argument is treated as one generator, not as a list of generators. In the second form, the identity element id may be given as the second argument.

It is not checked whether the underlying multiplication is associative, use `MagmaWithOne` (35.2-2) and `IsAssociative` (35.4-7) if you want to check whether a magma-with-one is in fact a monoid.

##### 52.1-3 Submonoid

`‣ Submonoid`( M, gens ) | ( function )

`‣ SubmonoidNC`( M, gens ) | ( function )

are just synonyms of `SubmagmaWithOne` (35.2-8) and `SubmagmaWithOneNC` (35.2-8), respectively.

##### 52.1-4 MonoidByGenerators

`‣ MonoidByGenerators`( gens[, one] ) | ( operation )

is the underlying operation of `Monoid` (52.1-2).

##### 52.1-5 AsMonoid

`‣ AsMonoid`( C ) | ( attribute )

If C is a collection whose elements form a monoid (see `IsMonoid` (52.1-1)) then `AsMonoid` returns this monoid. Otherwise `fail` is returned.

##### 52.1-6 AsSubmonoid

`‣ AsSubmonoid`( D, C ) | ( operation )

Let D be a domain and C a collection. If C is a subset of D that forms a monoid then `AsSubmonoid` returns this monoid, with parent D. Otherwise `fail` is returned.

##### 52.1-7 GeneratorsOfMonoid

`‣ GeneratorsOfMonoid`( M ) | ( attribute )

Monoid generators of a monoid M are the same as magma-with-one generators (see `GeneratorsOfMagmaWithOne` (35.4-2)).

##### 52.1-8 TrivialSubmonoid

`‣ TrivialSubmonoid`( M ) | ( attribute )

is just a synonym for `TrivialSubmagmaWithOne` (35.4-14).

##### 52.1-9 FreeMonoid

`‣ FreeMonoid`( [wfilt, ]rank[, name] ) | ( function )

`‣ FreeMonoid`( [wfilt, ]name1, name2, ... ) | ( function )

`‣ FreeMonoid`( [wfilt, ]names ) | ( function )

`‣ FreeMonoid`( [wfilt, ]infinity, name, init ) | ( function )

Called with a positive integer rank, `FreeMonoid` returns a free monoid on rank generators. If the optional argument name is given then the generators are printed as name `1`, name `2` etc., that is, each name is the concatenation of the string name and an integer from `1` to range. The default for name is the string `"m"`.

Called in the second form, `FreeMonoid` returns a free monoid on as many generators as arguments, printed as name1, name2 etc.

Called in the third form, `FreeMonoid` returns a free monoid on as many generators as the length of the list names, the i-th generator being printed as names`[`i`]`.

Called in the fourth form, `FreeMonoid` returns a free monoid on infinitely many generators, where the first generators are printed by the names in the list init, and the other generators by name and an appended number.

If the extra argument wfilt is given, it must be either `IsSyllableWordsFamily` (37.6-6) or `IsLetterWordsFamily` (37.6-2) or `IsWLetterWordsFamily` (37.6-4) or `IsBLetterWordsFamily` (37.6-4). This filter then specifies the representation used for the elements of the free monoid (see 37.6). If no such filter is given, a letter representation is used.

Also see Chapter 52.

##### 52.1-10 MonoidByMultiplicationTable

`‣ MonoidByMultiplicationTable`( A ) | ( function )

returns the monoid whose multiplication is defined by the square matrix A (see `MagmaByMultiplicationTable` (35.3-1)) if such a monoid exists. Otherwise `fail` is returned.

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